第3章：高等量子力学(学校讲义).pdf

Chapter 1. Hilbert Space §1. Linear Vector Spaces Define a linear vector space as follows. The operation “+” on a set of elements S = {x, y, z, . . .} is to have the following 4 properties: 1) {x ∈ S, y ∈ S} ⇒ z ≡ x + y ∈ S ∀ x, y ∈ S ; 2) The operation “+” is commutative and associative; x + y = y + x, (x + y) + z = x + (y + z) = x + y + z. 3) ∃ a unique element 0 x + 0 = x ∀ x ∈ S ; 4) ∀ x ∈ S ∃ − x x + (−x) = 0. This is, all the elements of S form an abelian group with respect to “+” operation. Furthermore, if the quantities α, β , γ , . . . are the elements of some field F, that we simply call scalars, then we require that multiplication of an element of S by a scalar satisfy (∀ x, y ∈ S , ∀ α, β ∈ F ): 5) α(βx) = (αβ)x; 6) (α + β )x = αx + βx; 7) α(x + y) = αx + αy ; 8) e · x = x (e is the unit element in F). Definition: any space S that is closed under the operations of addition and of multiplication by scalars is called a linear vector space. All elements in S are called vectors. Some examples of linear vector spaces: 1. n-dimensional Euclidean space E , n x = (ξ , ξ , . . . , ξ ), 1 2 n y = (η , η , . . . , η ), 1 2 n x + y = (ξ + η , ξ + η , . . . , ξ + η ), 1 1 2 2 n n αx = (αξ , αξ , . . . , αξ ). 1 2 n The vector space is real or complex according to whether the components of the vectors are real or complex. We hereafter take our field F of scalars to be t